Eventually, in order to get the best possible constraints on inflation
one will want to circumvent the slow-roll approximation completely, and
this can be done by computing the power spectra (first the scalar and
tensor spectra, and from them the induced microwave anisotropies) entirely
numerically.
Such an approach was recently described by Grivell and Liddle
[30],
and represents the optimal way to obtain constraints on
inflation from the data (though at present it has only been implemented
for single-field models).

Figure 1. The
4
potential as seen by
the PLANCK satellite. In the upper panel, the dashed
line shows the true
potential, and the full lines show a series of Monte Carlo reconstructions,
which differ in the realization of the observational errors. In reality
we get only one of these curves. The lower panel shows the combination
V'/V3/2, which is much better
determined. Scalar field values when scales
equalled the Hubble radius during inflation are shown, roughly
corresponding to the range of PLANCK.

In this approach, rather than estimating quantities such as the spectral
index n from the observations, one directly estimates the potential,
in some parametrization such as the coefficients of a Taylor series.
An example is shown in Figure 1, which shows a
test case of a 4
potential as it might be reconstructed by the PLANCK
satellite - see Ref.
[30]
for full details. Twenty different
reconstructions are shown (corresponding to
different realizations of the random observational errors), whereas in
the real
world we would get only one of these. We see considerable variation, which
arises because the overall amplitude can only be fixed by detection of the
tensor component, which is quite marginal in this model. However, there are
functions of the potential which are quite well determined. The lower panel
shows
V'/V3/2 (where the prime is a derivative with
respect to the field),
which is given to an accuracy of a few percent on the scales where the
observations are most efficient. This particular combination is favoured
because it is the combination which (at least in the slow-roll
approximation) gives the density perturbation spectrum.

No doubt, when first confronted with quality data people will aim to determine
n, r, and so on along with the cosmological parameters
such as the density
and Hubble parameter. However, if we become convinced that the inflationary
explanation is a good one, this direct reconstruction approach takes maximum
advantage of the data in constraining the inflaton potential.